Problem: Simplify and expand the following expression: $ \dfrac{6}{5q + 1}-\dfrac{2q + 2}{q + 3} $
In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(5q + 1)(q + 3)$ Multiply the first term by $\dfrac{q + 3}{q + 3}$ $ \begin{align*} \dfrac{6}{5q + 1} \times \dfrac{q + 3}{q + 3} & = \dfrac{(6)(q + 3)}{(5q + 1)(q + 3)} \\ & = \dfrac{6q + 18}{(5q + 1)(q + 3)}\end{align*} $ Multiply the second term by $\dfrac{5q + 1}{5q + 1}$ $ \begin{align*} \dfrac{2q + 2}{q + 3} \times \dfrac{5q + 1}{5q + 1} & = \dfrac{(2q + 2)(5q + 1)}{(q + 3)(5q + 1)} \\ & = \dfrac{10q^2 + 12q + 2}{(q + 3)(5q + 1)}\end{align*} $ Now we have: $ = \dfrac{6q + 18}{(5q + 1)(q + 3)} - \dfrac{10q^2 + 12q + 2}{(q + 3)(5q + 1)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{6q + 18 - (10q^2 + 12q + 2)}{(5q + 1)(q + 3)} $ $ = \dfrac{6q + 18 - 10q^2 - 12q - 2}{(5q + 1)(q + 3)} $ $ = \dfrac{-6q + 16 - 10q^2}{(5q + 1)(q + 3)}$ Expand the denominator: $ = \dfrac{-6q + 16 - 10q^2}{5q^2 + 16q + 3}$